I have tried evaluating this series 

$$\sum_{n=1}^{\infty}\frac{H_{n}^3}{(n+1)2^n} $$ 

using some methods but it's seems to me that it is very hard. However, I noticed that the series converges faster than the Riemann series.

**My question here is:**

Is there some mathematical technique for evaluating the above series?

**Note1:** Here, $H_n$ denotes the harmonic numbers.
 

**Edit** : I have a wrong type I meant in the denomenator $2^n$

Thank you for any help.